Reid's geography conjecture for surfaces of general type

Let $g=2,3,\ldots$. A smooth surface $X$ of general type has canonical self-intersection $K_X^2$ and holomorphic Euler characteristic $\chi(X,\mathcal{O}_X)$. A pencil of curves of genus at most $g$ means a fibration induced by a pencil whose general member has genus at most $g$. Reid's geography conjecture. There exist rational numbers $a_g$ and $b_g$ such that

$$a_2<a_3<\cdots,$$

and

$$\lim\limits_{g\to\infty}a_g=4,$$

with the property that, for every smooth surface $X$ of general type,

$$K_X^2\leq a_g\chi(X,\mathcal{O}_X)-b_g$$

if and only if $X$ has a pencil of curves of genus at most $g$. This conjecture predicts that sufficiently low points in the geography of surfaces of general type arise from surfaces admitting pencils of bounded genus; the source gives no resolution, so its status remains open.